Question

For S and S' in standard configuration, the Galilean transformations are:

x' = x - vt, y' = y, z' = z, t' = t

From the Lorentz transformations for v << c:

x' = x - vt, y' = y, z' = z, t' = t - vx/c^2

So it looks as if the Galilean transformations become increasingly accurate for:

vx -> 0, v << c

And exact for v = 0 for all x.

Yet, all text books I've come across state that the Galilean transformatons become more accurate for the condition v << c only.

So what are the conditions under which the Galilean transformations become more accurate and why?

Answer 1

If I understand correctly, the question is about whether, in deriving the Galilean transformation as an approximate limiting case of the Lorentz transformation, it is necessary to impose the requirement that is small, in addition to the "obvious" requirement that is small. The answer is yes, it is necessary. Of course, since is not a dimensionless quantity, we have to specify what we mean when requiring that it is small. The specific requirement is that , where ?t is the precision with which we wish to calculate time intervals.

Suppose that, in the inertial reference frame in which you are at rest, two stars explode simultaneously, one here and one in the Andromeda galaxy. Consider the same two events in the reference frame of someone walking at a liesurely pace. The Lorentz transformation indicates that they will be separated by a time interval of about 2 days. If you care about levels of precision less than that, you can't use the Galilean transformation, even though is small.